Prof. dr. ir. C.J. van Duijn, Dean Department of Mathematics and Computer Science &
Prof.dr. Onno Boxma, Chairman of the Mathematics Division TU/e, Scientific Advisor QPA/EURANDOM

Mean-Field Interaction Models for Large TCP Networks

This presentation will review various dynamical interaction models allowing one to analyze the throughputs obtained by a large collection of TCP-controlled flows sharing many links and routers, from the sole knowledge of the network parameters (capacity, buffer sizes, topology) and of the characteristics of each flow (RTT, route, on-off structure etc.).

In the tail-drop link, persistent flow case, the mean-field limit can be described geometrically as a billiards in the Euclidean space. This billiards has as many dimensions as the number of flow classes and as many reflection facets as there are routers and links. For non-persistent flows with an on-off structure, TCP can induce a turbulence like phenomenon which translates into two possible stationary regimes for the mean-field limit.

The AQM-RED link case can also be investigated by such mean-field techniques and leads to transport-type PDEs. In the single link case, this allows one to determine in closed form the stationary distribution of the stationary throughputs obtained by the flows, both in the persistent and the on-off cases.

When aggregated, the traffic generated by these models exhibits TCP and network-induced fluctuations that might explain some of the statistical properties observed on real traces.

October 26, 2004 Signal-to-Interference-Ratio Cells of a Spatial Point Process

Abstract
Consider a marked point process of the Euclidean space, where the mark of a point is a positive random variable that represents its  "emission power". Assume that the power radiated from a point decays in some isotropic way with Euclidean distance.

Define the signal-to-interference-ratio (SIR) cell of point X to be the region of the space where the power received from X is larger than some linear function of the interference. In this definition, the interference at some location of the space is the sum of the powers received by this location from all points other than X.

In this lecture we will analyze a few basis stochastic geometry questions pertaining to such SIR cells, in the case where the marks representing powers are independent, like the volume and the shape of the typical cell or the properties of the coverage of the space by SIR cells. We will also discuss the relationship between SIR cells and classical objects of stochastic geometry like Voronoi tessellations, germ grain models and the Johnson-Mehl model.

Slides

December 14, 2004  Percolation and Connectivity in Mobile Ad Hoc Networks

February 8, 2005 Cross layer Optimization in Mobile Ad Hoc Networks

March 29, 2005  A spatial Erlang Formula and its application to the loss probability in large CDMA networks

June 8, 2005 The Radial Spanning Tree (Joint work with C. Bordenave)

Abstract
In this talk, we analyze a class of spatial random spanning trees built on a realization of an homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of its edges or the mean degree of its nodes. Far away from the origin, these local properties are shown to be close to those of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some non local properties of the tree such as the shape and structure of its semi-infinite paths or the shape of the set of its nodes less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and in particular in wireless sensor communication networks where information has to be gathered at a central node.